1

Max-Min SINR in Large-Scale Single-Cell MU-MIMO:Asymptotic Analysis and Low Complexity Transceivers

Houssem Sifaou, Student Member, IEEE, Abla Kammoun, Member, IEEE, Luca Sanguinetti, SeniorMember, IEEE, Merouane Debbah, Fellow, IEEE and Mohamed-Slim Alouini, Fellow, IEEE

Abstract—This work focuses on the downlink and uplink oflarge-scale single-cell MU-MIMO systems in which the basestation (BS) endowed with M antennas communicates with Ksingle-antenna user equipments (UEs). Particularly, we aim atreducing the complexity of the linear precoder and receiver thatmaximize the minimum signal-to-interference-plus-noise ratiosubject to a given power constraint. To this end, we considerthe asymptotic regime in which M and K grow large witha given ratio. Tools from random matrix theory (RMT) arethen used to compute, in closed form, accurate approximationsfor the parameters of the optimal precoder and receiver, whenimperfect channel state information (modeled by the genericGauss-Markov formulation form) is available at the BS. Theasymptotic analysis allows us to derive the asymptotically optimallinear precoder and receiver that are characterized by a lowercomplexity (due to the dependence on the large scale componentsof the channel) and, possibly, by a better resilience to imperfectchannel state information. However, the implementation of bothis still challenging as it requires fast inversions of large matricesin every coherence period. To overcome this issue, we applythe truncated polynomial expansion (TPE) technique to theprecoding and receiving vector of each UE and make use ofRMT to determine the optimal weighting coefficients on a per-UE basis that asymptotically solve the max-min SINR problem.Numerical results are used to validate the asymptotic analysisin the finite system regime and to show that the proposed TPEtransceivers efficiently mimic the optimal ones, while requiringmuch lower computational complexity.

I. INTRODUCTION

Large-scale multiple-input multiple-output (MIMO) sys-tems, also known as massive MIMO systems, are consideredas a promising technique for next generations of wireless com-munication networks [1]–[4]. The massive MIMO technologyaims at evolving the conventional base stations (BSs) by usingarrays with a hundred or more small dipole antennas. Thisallows for coherent multi-user MIMO transmission where tensof users can be multiplexed in both uplink (UL) and downlink

L. Sanguinetti is with the University of Pisa, Dipartimento di Ingegneriadell’Informazione, Italy (luca.sanguinettiunipi.it) and also with the LargeSystems and Networks Group (LANEAS), CentraleSupelec, Universite Paris-Saclay, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France

M. Debbah is with the Large Systems and Networks Group (LANEAS),CentraleSupelec, Universite Paris-Saclay, 3 rue Joliot-Curie, 91192 Gif-sur-Yvette, France (m.debbah@centralesupelec.fr) and also with the Mathematicaland Algorithmic Sciences Lab, Huawei Technologies Co. Ltd., France (mer-ouane.debbah@huawei.com).

H. Sifaou, A. Kammoun and M.S. Alouini are with the Computer,Electrical, and Mathematical Sciences and Engineering (CEMSE)Division, KAUST, Thuwal, Makkah Province, Saudi Arabia (e-mail: houssem.sifaou@kaust.edu.sa, abla.kammoun@kaust.edu.sa,slim.alouini@kaust.edu.sa)

This research have been partially supported by the ERC Starting Grant305123 MORE, and by the research project 5GIOTTO funded by the Univer-sity of Pisa.

A preliminary version of this paper has been accepted in the IEEE Interna-tional Workshop on Signal Processing Advances in Wireless Communications,Edinburgh, UK, 2016.

(DL) of each cell. It is worth observing that, contrary to whatthe name “massive” suggests, massive MIMO arrays are rathercompact; 160 dual-polarized antennas at 3.7 GHz fit into theform factor of a flat-screen television [5].

The problem of designing precoder and receiver techniquesfor massive MIMO systems is receiving a lot of attention.Among the different optimization criteria, we distinguish thetransmit power minimization [6]–[8] and the maximizationof the minimum SINR [9], [10]. The latter is the focus ofthis work. In particular, we consider the case of a single-celllarge-scale multi-user (MU) MIMO system in which the BSmakes use of M antennas in order to communicate with Ksingle-antenna user equipments (UEs). Under the assumptionof perfect channel state information (CSI) at the BS, it isshown in [9] that the optimal linear precoder (OLP) for themax-min SINR problem is closely related to the optimallinear receiver (OLR), as it can be computed by exploitingthe UL-DL duality principle. The latter allows to convert theDL optimization problem into its equivalent counterpart inthe dual UL variables. The OLP is then found in the formof a fixed-point problem whose solution corresponds to thepowers allocated to the UEs in the dual UL network. Althoughcomputationally feasible, the above approach does not provideany insight into the structure of both OLP and OLR.

To solve the above issue, we follow the same approach asin recent works [11]–[14] (among many others). Particularly,we consider the asymptotic regime in which M and K growlarge with bounded ratio, which allows us to leverage recentresults from random matrix theory. The analysis is performedunder the assumption of imperfect CSI at the BS, which ismodeled by the generic Gauss-Markov formulation form (seefor example [15]). Under imperfect CSI, the OLP and OLRderived in [9] are no longer optimal [16]. This is clearly un-veiled by the large system analysis, which additionally showsthat the directions of the precoding and receiving vectors aswell as their associated powers converge asymptotically todeterministic values depending only on the long-term channelattenuations of the UEs. In order to account for the channelestimation errors and to avoid the need for solving fixed pointequations at the pace of fast fading channels, we propose theasymptotically OLP and OLR (called A-OLP and A-OLR,respectively) for which the same asymptotic directions as OLPand OLR are used but the transmit powers are computed inorder to maximize the asymptotic minimum SINR under atotal power constraint. We prove that A-OLP provides asymp-totically better performance than OLP while OLR and A-OLRexhibit the same performance in the asymptotic regime.

Despite being reduced compared to OLP and OLR, the im-plementation of A-OLP and A-OLR might be of prohibitivelyhigh complexity in large scale MIMO systems due to the need

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for computing the inverse of large matrices, whose dimensionsgrow with M and K. To tackle this problem, we resort tothe truncated polynomial expansion (TPE) technique, whichhas recently been applied to reduce the complexity of theRZF precoder in [17], [18] and the MMSE receiver in [19]–[22]. In all these aforementioned works, the TPE concept isapplied using the same weighting coefficients for all UEs.This limits the number of degrees of freedom with an ensuingdegradation of the maximum achievable performance. In lightof this observation, we employ the TPE technique on a per-UE basis. More specifically, the TPE concept is applied toeach vector of the precoding and receiving matrices ratherthan to the whole matrices themselves. This leads to the so-called user specific TPE (US-TPE) precoder and receiver forwhich approximations of the resulting SINRs are computedthrough asymptotic analysis. These results are then used tooptimize the US-TPE parameters in order to maximize theminimum SINR over all UEs in the DL and UL. Interestingly,the optimization problem can be cast in both cases as the max-min SINR problems previously studied in [7], [9], [10]. Thesolution of such problem leads to a novel US-TPE precoderand receiver, which are shown by simulations to achievealmost the same performance as A-OLP and A-OLR, whilerequiring much lower computational complexity.

The remainder of this work is organized as follows. Nextsection introduces the system model and formulates the max-min SINR problem for both DL and UL. Section III dealswith the large system analysis of OLP and OLR as well aswith the design of both under the assumption of imperfectCSI. The proposed TPE precoder and receiver are presentedin section IV. Numerical results are shown in Section V whilesome conclusions are drawn in Section VI.

Notations – Boldface lower case is used for denoting columnvectors, x, and upper case for matrices, X, XT , XH denotethe transpose and conjugate of X, respectively. The trace ofa matrix X is denoted by tr(X). A circularly symmetriccomplex Gaussian random vector x is denoted x ∼ CN (x,Q)where x is the mean and Q is the covariance matrix. Moreover,IM denotes the M ×M identity matrix and 0M×1 stands forthe M × 1 vector with all entries equal to zero. The expec-tation operator is denoted E[.]. For an infinitely differentiablefunction f(t), the n-th derivative at t = t0 is denoted f (n)(t0)and it is simply denoted by f (n) when t = 0. The operatordiag

({vk}Kk=1

)is the diagonal matrix having v1, · · · , vK as

diagonal elements.

II. SYSTEM MODEL AND PROBLEM FORMULATION

We consider the DL and UL of a single-cell multi-userMIMO system in which the BS is equipped with M antennasand communicates with K < M single antenna UEs. TheK active UEs are randomly selected from a large set ofUEs within the coverage area. We denote by hk ∈ CMthe channel vector of UE k and assume that hk =

√βkzk

where zk ∼ CN (0, IM ) is the small-scale fading channeland βk accounts for the corresponding large-scale channelfading or path loss. Within the above setting, we are in-terested in computing the optimal linear precoder (receiver)

that maximizes the minimum SINR in the DL (UL) whilesatisfying a total average power constraint Pmax. Under theassumption of perfect CSI at the BS, the solution of thisproblem is well known and can be computed using differentapproaches based on standard convex optimization techniques.Next, for completeness we consider the DL and UL andreview the optimal linear precoder and receiver structure. Thiswill be instrumental for the asymptotic analysis performedsubsequently.

A. Downlink

Denoting by gk ∈ CM the precoding vector associated withUE k, the signal received at UE k can be written as

yk = hHk gksk +K∑

i=1,i6=khHk gisi + nk (1)

where si ∼ CN (0, 1) is the signal intended to UE k, assumedindependent across k, nk ∼ CN (0, 1/ρ) accounts for theadditive Gaussian noise with ρ being the effective signal-to-noise ratio (SNR). The DL SINR at the k-th UE is:

SINRdlk =

|hHk gk|2K∑

i=1,i6=k|hHk gi|2 + 1/ρ

(2)

and the total average transmit power per UE is 1K tr(GGH)

where G = [g1, · · · ,gK ] ∈ CN×K . The latter is chosen asthe solution of the following max-min SINR problem:

Pdl :

{maxG

mink

SINRdlk

γk

s.t. 1K tr(GGH) ≤ Pmax

(3)

where γk is a factor reflecting the priority of UE k and Pmax

is the power constraint at the BS. In [9], [16], it is shown thatthe column vectors of the optimal linear precoder (OLP) G?

solving Pdl take the form g?k =√

p?kK

v?k||v?k||

with

v?k =

K∑

`=1, 6=k

q?`K

h`hH` +

1

ρIM

−1

hk (4)

where the scalars {q?k} are obtained as the unique positivesolution to the following fixed-point equations:

q?k =γkτ

?

1KhHk

(K∑

k=1, 6=k

q?`K h`hH` + 1

ρIM

)−1

hk

(5)

with τ? being the minimum weighted SINR given by [9]:

τ? =KPmax

K∑n=1

γn

1KhHn

(K∑

k=1,k 6=n

q?kK hkhHk + 1

ρIM

)−1

hn

−1 . (6)

The optimal power coefficients {p?k} are such that the follow-ing equalities are satisfied [9]:

SINRdl?1

γ1= · · · = SINRdl?

K

γK= τ? (7)

3

with

SINRdl?

k =

p?kK|hHk v?k|2||v?k||2

K∑i=1,i6=k

p?iK

|hHk v?i |2||v?i ||2

+ 1/ρ

. (8)

From the above condition, it turns out that p? = [p?1, · · · , p?K ]T

can be obtained as [9]:

p? =τ?

ρ(IK − τ?ΓF)

−1Γ1K (9)

where Γ = diag{Kγ1‖v?1‖2|hH1 v?1 |2

, · · · , KγK‖v?K‖2

|hHKv?K |2}

and F ∈ CK×Khas elements given by:

[F]k,i =

{0 if k = i1K|hHk v?i |2‖v?i ‖2

if k 6= i.(10)

B. Uplink

From the UL-DL duality shown in [9], it follows that thevectors {v?k} and q? = [q?1 , · · · , q?K ]

T can be obtained as thesolution of the following uplink max-min SINR problem:

Pul :

{max{vk},q

mink

SINRulk

γk

s.t. 1K1TKq ≤ Pmax

(11)

with

SINRulk =

qkK |hHk vk|2

vHk

(K∑

i=1,i6=kqiKhihHi + 1

ρIM

)vk

. (12)

From (12), it easily follows that the vector vk solving Pul

coincides with the minimum-mean-square-error (MMSE) re-ceiver [23]. Next, we refer to the solution of Pul as the optimallinear receiver (OLR).

III. LARGE SYSTEM ANALYSIS

As shown above, the OLP and OLR are parametrized by thescalars {q?k} and {p?k} where {q?k} need to be evaluated by aniterative procedure due to the fixed-point equations in (5) and(6). This is a computationally demanding task when M andK are large since the matrix inversion operation in (5) and (6)must be recomputed at every iteration. Moreover, computing{q?k} as the fixed point of (5) and (6) does not provide anyinsight into the optimal structure of {q?k} and consequently of{p?k} in (9). In addition, both depend directly on the channelvectors {hk} and change at the same pace as the small-scalefading (i.e., at the order of milliseconds). To overcome theseissues, we exploit the statistical distribution of {hk} and thelarge values of M,K (as envisioned in future networks) tocompute deterministic approximations (also known as deter-ministic equivalents) of {q?k} and {p?k}. For technical purposes,we shall consider the following assumptions:

Assumption 1. We assume that both M and K grow large,their ratio being bounded below and above as follows: 1 <lim inf MK ≤ lim sup M

K <∞.Assumption 2. The channel attenuation coefficients {βk}satisfy: 0 < lim inf {βk} ≤ lim sup {βk} <∞.

Assumption 3. The power coefficients {pk} satisfy: 0 <lim inf mini pi < lim sup maxi pi <∞.

We also assume the BS has imperfect knowledge of theinstantaneous channel realizations {hk}. As in many otherworks [24], [15], [25], this is modeled by the generic Gauss-Markov formulation form ∀k:

hk =√βk(√

1− η2zk + ηzk) (13)

=√

1− η2hk +√βkηwk (14)

where wk ∼ CN (0, IM ) accounts for the channel estimationerrors independent of the fast fading channel vector zk. Thescalar parameter η ∈ [0, 1] indicates the quality of the instan-taneous CSI, i.e., η = 0 corresponds to perfect instantaneousCSI and η = 1 corresponds to having only statistical chan-nel knowledge.1 The matrix collecting the estimated channelvectors is denoted H = [h1, · · · , hK ].

When only imperfect CSI is available at the BS, the struc-ture of the OLP and OLR is not known (most of the existingsolutions in the literature are based on heuristic approaches).To overcome this issue, we assume that the true channels{hk} are simply replaced by their estimates {hk} (which isan accurate procedure for good CSI quality). This yields

gk =

√pkK

vk‖vk‖

(15)

where vk =(∑K

`=1, 6=kq`K h`h

H` + 1

ρIM

)−1

hk and the coef-ficients {qk} are obtained as:

qk =γk τ

1K hHk

(K∑

`=1, 6=kq`K h`hH` + 1

ρIM

)−1

hk

∀k (16)

with

τ =KPmax

K∑n=1

γn

1K hHn

(K∑

k=1,k 6=nqkK hkhHk + 1

ρIM

)−1

hn

−1 . (17)

The transmit powers {pk} are such that (9) is satisfied afterreplacing {hk} with {hk} and τ? with τ . Next, we resortto the large dimensional analysis and show that qk and pkgets asymptotically close to explicit deterministic quantities asM and K grow large as for Assumption 1. These quantitiesprovide some insights on the structure of the precoder andreceiver as well as on how the different parameters (suchas large scale channel gains, imperfect channel knowledge,UE priorities, maximum transmit power) affect the systemperformance.

A. Asymptotic analysis of OLP and OLR under imperfect CSI

Our first result is as follows:

1Observe that the same η is assumed for all UEs only for simplicity. Thegeneralization to different η’s is straightforward.

4

Theorem 1. Under the settings of Assumptions 1 and 2, wehave that maxk |τ − τ | → 0 where τ is the unique positivesolution to the following fixed point equation:

τ =ρPmax

1K

K∑i=1

γiβi

(M

K− 1

K

K∑

i=1

γiτ

1 + γiτ

). (18)

Also, we have that maxk |qk − qk| → 0 where

qk =γkβk

Pmax

1K

K∑i=1

γiβi

. (19)

Proof: The proof relies on the observation that all thequantities dk , γk

βkτqk

should converge to the same determin-istic limit. Note that to determine this limit, standard tools fromrandom matrix theory cannot be applied since {dk} depends onthe channel vectors {hk} in a non-linear fashion. To overcomethis issue, we make use of the techniques developed recentlyin [26]. Details are provided in Appendix B.

The above theorem provides the explicit form of {qk},whose computation requires only knowledge of the UEs prior-ity coefficients {γk} and the channel attenuation coefficients{βk}. The latter can be easily estimated since they changeslowly with time. Observe that in the DL the parameter q?kis known to act as a UE priority parameter that implicitlydetermines how much interference a specific UE k may induceto the other UEs in the cell [16]. Interestingly, its asymptoticvalue qk is proportional to γk and inversely proportional toβk. Higher priority is thus given to UEs that require high per-formance (large γk) and/or have weak propagation conditions(small βk). In the UL, q?k corresponds to the transmit powerof UE k. Consequently, (19) indicates that in the asymptoticregime more power is given to UEs with higher priorities andweaker channel conditions.

The asymptotic transmit powers in DL are given in explicitform as follows:

Theorem 2. Under the settings of Assumptions 1 and 2, wehave maxk |pk − pk| → 0 where

pk =γkβk

τ

ξ

(βkPmax

(1 + γkτ)2+

1

ρ

)(20)

and ξ is positive and given by

ξ =M

K− 1

K

K∑

i=1

(γiτ)2

(1 + γiτ)2. (21)

Proof: The proof of the convergence of {pk} followsalong the same arguments as those used for {qk}, and it isthus omitted for space limitations.

The results of Theorems 1 and 2 can be used to compute anasymptotic expression of the SINRs in DL and UL as providedby the following lemmas:

Lemma 3. Under the settings of Assumptions 1 and 2, wehave maxk |SINRdl?

k − SINRdl

k | → 0 where

SINRdl

k =pk(1− η2)ξ

µkPmax + 1ρβk

(22)

with

µk =1 + 2η2γkτ + η2γ2

kτ2

(1 + γkτ)2(23)

Proof: By using standard calculus from random matrixtheory, it is easily seen that the asymptotic expression of{SINRdl

k } remains almost surely the same if pk and qk arereplaced by pk and qk. Then, using similar techniques asthose in [13], [17], deterministic equivalents of the signal andinterference terms can be computed, leading thus to (22). SeeAppendix C for details.

Lemma 4. Under the settings of Assumptions 1 and 2, wehave maxk |SINRul?

k − SINRul

k | → 0 where

SINRul

k =qk(1− η2)ξ

1K

K∑i=1

βiβkµiqi + 1

ρβk

. (24)

Proof: The proof relies on the same techniques used inAppendix C and it is thus omitted.

An important consequence of the above results is thatthe performance of the network in DL and UL remainsasymptotically the same if {qk} and {pk} are replaced with{qk} and {pk} such that the precoding/receiving vector of UEk is computed as:

gk =

√pkK

vk‖vk‖

(25)

with

vk =

(K∑

i=1

qiK

hihHi +

1

ρIM

)−1

hk. (26)

This result is particularly interesting from an implementationpoint of view. Indeed, unlike {qk} and {pk}, {qk} and {pk}in (19) and (20) depend only on the large-scale channelstatistics. As a consequence, {qk} and {pk} are not requiredto be computed at every channel realization but only onceper coherence period. This provides a substantial reductionin computational complexity as compared to OLP and OLRsince solving (5) and (6) at the pace of fast fading channel isno longer required.

The above analysis reveals also that the asymptotic SINRs in(22) and (24) are both decreasing functions of η with maximalvalue achieved for η = 0 and given by

SINRdl

k,max

γk=

SINRul

k,max

γk= τ (27)

from which it follows that:

Corollary 1. If perfect CSI is available, then in the asymptoticregime the minimum weighted SINR is the same for both DLand UL.

Unlike the SINR expressions, the coefficients qk and pk in(19) and (20) are found to be independent of η. This is due tothe fact that they depend solely on the statistics of estimatedchannel vectors hk, which are the same of the true channel

5

vectors hk regardless of the value of η.2 Next, we follow adifferent approach, which aims at designing the OLP and OLRby exploiting the above large system analysis. As shown next,the idea is to still use the vectors vk in (26) but to design theDL and UL transmit powers so as to maximize the asymptoticminimum SINR.

B. Asymptotic design of OLP and OLR with imperfect CSI

To begin with, let us call p = [p1, . . . , pK ]T and q =[q1, . . . , qK ]T the DL and UL power vectors, respectively, andassume that they are kept fixed. Assume also that the precodingvectors are computed as

gk =

√pkK

vk‖vk‖

(28)

with vk given by (26). Therefore, a direct application of

Lemma 3 yields maxk |SINRdlk (p)− SINR

dl

k (p)| → 0 with

SINRdl

k (p) =pk(1− η2)ξ

µkK

K∑i=1

pi + 1ρβk

. (29)

Accordingly, from Lemma 4 it follows that

maxk |SINRulk (q)− SINR

ul

k (q)| → 0 where

SINRul

k (q) =qk(1− η2)ξ

1K

K∑i=1

βiβkµiqi + 1

ρβk

. (30)

The main contribution of this section unfolds from the aboveresults and provides the DL and UL power vectors p and qthat maximize the asymptotic minimum SINR in DL and ULunder the power constraint Pmax. This amounts to solving thefollowing optimization problems:

PAdl :

maxp

mink

SINRdl

k (p)γk

s.t. 1K1TK p ≤ Pmax

(31)

and

PAul :

maxq

mink

SINRul

k (q)γk

s.t. 1K1TK q ≤ Pmax.

(32)

Define the diagonal matrix D ∈ CK×K

D = diag

(γ1

ξβ1(1− η2), · · · , γK

ξβK(1− η2)

)(33)

and the vector f ∈ CK with entries given by [f ]i = βiµi/K.Using the above notation, PAdl and PAul can be rewritten as

PAdl :

{maxp

mink

pk

[D(f1T p+ 1ρ1)]

k

s.t. 1K1TK p ≤ Pmax

(34)

and

PAul :

{maxq

mink

qk

[D(1fT q+ 1ρ1)]

k

s.t. 1K1TK q ≤ Pmax

(35)

2Observe that E{hk} = E{hk} = 0, E{hkhHk } = E{hkh

Hk } = βkIM

and E{hihHk } = E{hih

Hk } = 0M .

Following [9], it can be easily shown that PAdl and PAul arerelated by the UL-DL duality. Therefore, from [9], [10] itfollows that the optimal power vectors p? and q? are suchthat:

p? ∝ D

(f1T +

1

ρKPmax11T

)p? (36)

and

q? ∝ D

(1fT +

1

ρKPmax11T

)q?. (37)

As seen, p? and q? are proportional to the Perron eigenvectors[27] of the non negative matrices D(f1T + 1

ρKPmax11T )

and D(1fT + 1ρKPmax

11T ) respectively. Using the inequalityconstraints, we finally obtain:

p? =KPmax

1TD(f + 1ρKPmax

1)D(f +

1

ρKPmax1) (38)

and

q? =KPmax

1TD1D1. (39)

In a more explicit form, we have that:

p?k =Pmaxγkµk + γk

ρβk

1K

K∑i=1

γiµi + γiρβiPmax

(40)

and

q?k =γkβk

Pmax

1K

K∑i=1

γiβi

. (41)

Unlike {pk} in (20), the DL powers {p?k} depend on thechannel estimation accuracy through {µi}. This makes the so-called asymptotic OLP (A-OLP) achieve better performancethan OLP as shown later by simulations. On the other hand,the UL powers {q?k} coincide with {qk} in (19), obtained bycomputing the deterministic equivalents of {qk}. This is dueto the fact that both solutions rely on the same beamformingreceive directions vk. Therefore, the asymptotic OLR (A-OLR) is identical to OLR.

As mentioned before for OLP and OLR, the use of {q?k}and {p?k} largely simplifies the implementation of A-OLP andA-OLR as their computation requires only knowledge of thelarge scale channel statistics and must be performed only onceper coherence period (rather than at the same pace as the small-scale fading). Despite being simplified, the implementation ofA-OLP and A-OLR as well as that of OLP and OLR stillrequires the matrix inversion operation in (26). This can be atask of a prohibitively high complexity when M and K arelarge as envisioned in large scale MIMO systems. To addressthis issue, a TPE approach will be adopted next.

IV. USER SPECIFIC TPE PRECODING AND RECEIVER

The common way to apply the TPE concept consists inreplacing the matrix inverse by a weighted matrix polynomialwith J terms [17], [28]. Differently from the traditionalapproach, we propose in this work to apply the truncation

6

artifice separately to each vector of the precoding and receivingmatrices.

Applying the TPE on a per-UE basis, the precoding vectorassociated with UE k writes as:

gdlk,TPE =

√pk,TPE

K

vk,TPE

‖vk,TPE‖(42)

with

vk,TPE =

J−1∑

`=0

wdl`,k

(HQHH

K

)`hk√K

(43)

where J is the truncation order and Q = diag(q1, · · · , qK).Plugging gdl

k,TPE into (2) and letting wk,dl =

[wdl0,k, · · · , wdl

J−1,k]T , the SINR corresponding to UE kcan be written as:

SINRdlk,TPE =

pk,TPEwHk,dlaka

Hk wk,dl

wHk,dlEkwk,dl

∑i6=k

pi,TPE

K

wHi,dlBk,iwi,dl

wHi,dlEiwi,dl+ 1

ρ

(44)

where ak ∈ CJ×1, bk ∈ CJ×1, and Bi,k ∈ CJ×J are givenby:

[ak]` =1

KhHk

(HQHH

K

)`hk (45)

[Bk,i]`,m =1

KhHk

(HQHH

K

)`hih

Hi

(HQHH

K

)mhk (46)

[Ek]`,m =1

KhHk

(HQHH

K

)`+mhk. (47)

The transmit power at the BS can be easily found as P =1K

∑Kk=1 pk,TPE.

The TPE concept is now applied in UL to the OLR. Let{ qk,TPE

K

}be the set of UL transmit powers. The receive

beamforming vector associated with UE k is thus given by:

gulk,TPE =

J−1∑

`=0

wul`,k

(HQHH

K

)hk√K. (48)

Plugging gulk,TPE into (12) yields the SINR of UE k given by:

SINRulk,TPE =

qk,TPEwHk,ulaka

Hk wk,ul∑

i 6=k

qi,TPE

K wHk,ulBi,kwk,ul + 1

ρwHk,ulEkwk,ul

(49)where wk,ul =

[wul

0,k, · · · , wulJ−1,k

]Tand ak, Bi,k Ek are

given by (45) – (47).

V. ASYMPTOTIC ANALYSIS AND OPTIMIZATION OF THEUSER SPECIFIC TPE PRECODER AND RECEIVER

In this section, we consider the asymptotic regime definedin Assumption 1 and show that the SINRs of the TPE precoderand receiver converge to deterministic equivalents, that dependonly on the weighting vectors {wk,dl}Kk=1 or {wk,ul}Kk=1,and the large-scale channel statistics. These deterministicequivalents are then exploited to compute the optimal weightsthat maximize the minimum asymptotic DL/UL SINR.

A. Asymptotic Analysis

Let us introduce the fundamental equations that are neededto express the deterministic equivalents. We begin by definingδ(t) as the unique positive solution of the following equation∀t > 0:

δ(t) =M

K

1

1 + tK

K∑i=1

βiqi1+tδ(t)qiβi

. (51)

Define Xk(t) and Zk,i(t) as:

Xk(t) =βkδ(t)

1 + tqkβkδ(t), (52)

Zk,i(t, u) =βifk(t, u)α(t, u)

(1 + tδ(t)βiqi)(1 + uδ(u)βiqi)(53)

with fk(t, u) being given by:

fk(t, u) = βk

(η2 +

1− η2

(1 + qkβktδ(t))(1 + qkβkuδ(u))

)(54)

and

α(t, u) =δ(t)δ(u)

MK − tu

K δ(t)δ(u)K∑i=1

[βiqi]2

[1+tqiβiδ(t)][1+uqiβiδ(u)]

.

(55)Let ak ∈ CJ be defined as:

[ak]` =(−1)`

`!

√1− τ2X

(`)

k (56)

and call Bi,k ∈ CJ×J and Ek ∈ CJ×J the matrices withelements given by:

[Bk,i

]`,m

=(−1)`+m

`!m!Z

(`+m)

k,i (57)

[Ek

]`,m

=(−1)`+m

(`+m)!X

(`+m)

k . (58)

The main technical result of this section then lies in thefollowing lemma:

Lemma 5. Under the settings of Assumptions 1 and2, we have maxk |SINRdl

k,TPE − SINRdl

k,TPE| → 0 and

maxk |SINRulk,TPE − SINR

ul

k,TPE| → 0 with

SINRdl

k,TPE =pk,TPE

wHk,dlakaHk wk,dl

wHk,dlEkwk,dl

∑i 6=k

pi,TPE

K

wHi,dlBk,iwi,dl

wHi,dlEkwi,dl+ 1

ρ

(59)

SINRul

k,TPE =qk,TPEwH

k,ulakaHk wk,ul∑

i 6=k

qi,TPE

K wHk,ulBi,kwk,ul + 1

ρwHk,ulEkwk,ul

.

(60)

Also, we have that P − P → 0 with

P =1

K

K∑

k=1

wHk Ekwk. (61)

Proof: The deterministic equivalents of the SINR andtransmit powers are obtained by computing the asymptotic

7

q?k,TPE =γkKPmax

K∑`=1

γ`aTkE− 1

2k

(∑i6=k

ρK q

?i,TPEE

− 12

k Bi,kE− 1

2k +IJ

)−1

E− 1

2k ak

aT` E− 1

2`

(∑j 6=`

ρK q

?j,TPEE

− 12

` Bj,`E− 1

2` +IJ

)−1

E− 1

2` a`

∀k (50)

expressions of the entries of ak, Bk,i and Ek. The latter canbe written as a function of the derivatives of some quadraticforms whose deterministic equivalents are known in randommatrix theory. See Appendix D for details.

With the asymptotic equivalents of the SINR and thetransmit power on hand, we are ready now to determine theoptimal parameters of the TPE based receiver and precoder.

B. Optimization of the US-TPE precoding and receiver

In the sequel, we compute the optimal weighting vectorswk,dl and wk,ul as well as the optimal DL and UL transmitpowers. To begin with, we let

ck,dl =E

12

kwk,dl

‖E12

kwk,dl‖ck,ul =

E12

kwk,ul

‖E12

kwk,ul‖(62)

and rewrite the asymptotic SINR expressions in (59) and (60)as follows:

SINRdl

k,TPE =pk,TPE cHk,dlE

− 12

k akaHk E− 1

2

k ck,dl

∑i 6=k

pi,TPE

K cHi,dlE− 1

2

i Bk,iE− 1

2

i ci,dl + 1ρ

(63)

SINRul

k,TPE =qk,TPEcHk,ulE

− 12

k akaHk E− 1

2

k ck,ul

∑i 6=k

qi,TPE

K cHk,ulE− 1

2

k Bi,kE− 1

2

k ck,ul + 1ρ

. (64)

The parameters {ck,dl}, {ck,ul}, {pk,TPE} and {qk,TPE} arecomputed as solutions of the following optimization problems:

PTPEdl :

max{ck,dl},pTPE

mink

SINRdlk,TPE

γk

s.t. 1K1TKpTPE ≤ Pmax

(65)

and

PTPEul :

max{ck,ul},qTPE

mink

SINRulk,TPE

γk

s.t. 1K1TKqTPE ≤ Pmax

(66)

which have the same structure of (3) and (11). Followingsimilar arguments, it turns out that the solution is such that allthe weighted asymptotic SINRs are equal to τ?TPE:

τ?TPE =SINR

ul

k,TPE

γk=

SINRdl

k,TPE

γk∀k. (67)

The optimal values q?k,TPE are obtained as the unique solutionof the fixed-point system of equations in (50) whereas theoptimal weighting vectors are such that c?k,ul = c?k,dl = c?kwith:

c?k =

( ∑i 6=k

q?i,TPE

K E− 1

2

k Bi,kE− 1

2

k + 1ρIJ

)−1

E− 1

2

k ak

∥∥∥∥∥( ∑i6=k

q?i,TPE

K E− 1

2

k Bi,kE− 1

2

k + 1ρIJ

)−1

E− 1

2

k ak

∥∥∥∥∥

. (68)

0 0.2 0.4 0.6 0.8

0.5

1

1.5

CSI parameter ηA

vera

gepe

rU

Era

te[b

its/s

ec/H

z]

OLP

A-OLP

US-TPE precoder (J = 2)

US-TPE precoder (J = 1)

Fig. 1. Average per UE rate vs. η when K = 32, M = 128, Pmax = 5Watt and ρ = 20 dB.

From (50) , it follows that the computation of {q?k,TPE}requires matrix inversions whose complexity depend on J .However, J is small and does not need to scale with thevalues of M and K. Thus, the computation of q?k,TPE is notvery demanding. The optimal power vector p?TPE is such thatthe weighted SINRs in the uplink are all equal to τ?TPE. Thisyields:

pTPE =τ?TPE

ρ(IK − τ?TPEΓTPEFTPE)

−1ΓTPE1K (69)

where:

ΓTPE = diag

{(cHk,dlE

− 12

k akaHk E− 1

2

k ck,dl

)−1}K

k=1

(70)

and

[FTPE]k,i =

{0 if k = i1K cHi,dlE

− 12

k Bk,iE− 1

2

k ci,dl if k 6= i.(71)

From the above results, it follows that the TPE-based schemeshave the same structure as A-OLP and A-OLR. However, theformer allow a considerable reduction in the complexity sincethey require only about O(KM) arithmetic operations as theydo not involve the computation of a matrix inverse. This hasto be compared with the OLP and OLR that involve O(K2M)arithmetic operations. For more details about complexity anal-ysis and saving, we refer the reader to [17] where the benefitsof TPE when applied to precoding schemes are discussed indetails.

VI. SIMULATION RESULTS

Numerical results are now used to make comparisons amongthe different transceiver schemes and to validate the asymptotic

8

3 4 5 6 7 8 9 101

1.5

2

2.5

3

Power constraint Pmax [Watt]

Ave

rage

per

UE

rate

[bits

/sec

/Hz]

OLP

US-TPE precoder (J = 3)

US-TPE precoder (J = 2)

MRT (US-TPE precoder J = 1)

Fig. 2. Average per UE rate vs. power constraint Pmax when K = 32,M = 128, ρ = 20 dB and η = 0.

analysis. The UEs are assumed to be uniformly distributedin a cell with radius 250 m. The path loss βk between theBS and a UE k with distance xk from the BS is modeledas βk = 1/1 + (xk/d0)

δ where δ = 3.8, d0 = 30 m. Theanalysis is conducted in terms of the average achievable rateper UE given by:

r =1

K

K∑

k=1

E [log2(1 + SINRk)] (72)

where the expectation is taken with respect to the differentchannel realization. We set ρ = 20 dB and assume that theUEs’ priorities {γk} are randomly chosen from the interval[1, 2]. Markers are used to represent the asymptotic resultswhereas the error bars indicate the standard deviation of theMonte Carlo results.

Fig. 1 reports the downlink average rate per UE of OLP, A-OLP and US-TPE precoding as a function of η when K = 32,M = 128 and Pmax = 5 Watt. As seen, when η takessmall values, A-OLP and OLP have approximately the sa meperformances. As η increases, OLP presents a more significantloss in average per UE rate performances. Moreover, it canbe seen that US-TPE with J = 2 achieves almost the sameperformance as A-OLP and this over all the range of η.This clearly confirms that US-TPE shares the same interestingfeatures of A-OLP while requiring a lower complexity.

Fig. 2 investigates the DL average per UE rate with respectto the power budget Pmax when K = 32, M = 128, η = 0(perfect CSI case) and ρ = 20 dB. An important observationfrom Fig. 2 is that the gap in performance between US-TPEand OLP increases with Pmax. To reduce this gap, one solutionis to use the US-TPE with higher truncation orders.

A similar analysis is now conducted for the UL. Only OLRis considered since A-OLR and OLR have asymptotically thesame performances. Fig. 3 illustrates the uplink average rateper UE vs. η. As seen, with J = 2, US-TPE receiver providesthe same performance as OLR.

Fig. 4 illustrates the uplink average rate per UE vs. thenumber of BS antennas M for different values of Pmax

0 0.2 0.4 0.6 0.8

0.5

1

1.5

CSI parameter η

Ave

rage

per

UE

rate

[bits

/sec

/Hz]

OLR

US-TPE receiver (J = 2)

US-TPE receiver (J = 1)

Fig. 3. Average per UE rate vs. η when K = 32, M = 128, Pmax = 5Watt and ρ = 20 dB.

80 100 120 140 160 180

1

2

3

Pmax = 3

Pmax = 5

Pmax = 10

Number of BS antennas M

Ave

rage

per

UE

rate

[bits

/sec

/Hz]

OLR

US-TPE receiver (J = 2)

Fig. 4. Average per UE rate vs. M when K = 32, ρ = 20 dB and η = 0.

when η = 0. As seen, US-TPE receiver provides comparableperformance to OLR for all values of M . Besides, the gapincreases with Pmax as in DL, and seems to be weaklydependent of the number of antennas M .

VII. CONCLUSIONS

This work considered a single-cell large-scale MU-MIMOsystem and studied the problem of designing the optimallinear transceivers that maximize the minimum SINR whilesatisfying a certain power constraint. We considered theasymptotic regime in which the number of BS antennas Mand the number of the UEs K grow large with the samepace. Stating and proving new results from large-scale randommatrix theory allowed us to give concise approximations of theoptimal transceivers. Such approximations turned out to be ofmuch lower complexity as they depend only on the long-termchannel attenuations of the UEs, the maximum transmit powerand the quality of the channel estimates. Numerical resultsindicated that these approximations are very accurate even forsmall system dimensions. To further reduce the computationalcomplexity, we proposed to apply the truncated polynomial

9

expansion technique to the precoding and receiving vectors ofeach UE. The resulting transceiver was then optimized in theasymptotic regime. Numerical results showed that it achievesa-close-to-optimal performance.

APPENDIX AUSEFUL LEMMAS

This appendix gathers some technical results from randommatrix theory concerning the asymptotic behaviour of largerandom matrices. Next, we denote by X = [x1, · · · ,xK ] aM × K standard complex Gaussian matrix. Let t > 0 andR = diag (α1, · · · , αK). We define the resolvent matrix ofXRXH as:

Q(t) =

(t

K

K∑

i=1

αixixHi + IM

)−1

=

(t

KXRXH + IM

)−1

.

(73)Define also Qk(t) as:

Qk(t) =

t

K

∑

i6=kαixix

Hi + IM

−1

(74)

which is obtained from Q(t) by removing the contribution ofvector xk. The following lemmas recall some classical identi-ties involving the resolvent matrix, which will be extensivelyused in our derivations:

Lemma 6. The following identities hold true:1) Inverse of resolvents:

Q(t) = Qk(t)− tαkQk(t)xkxHk Qk(t)

1 + tαkK xHk Qk(t)xk

. (75)

2) Rank-one perturbation result: For any matrix A, we havetrA (Q(t)−Qk(t)) ≤ ‖A‖2.

Lemma 7 (Convergence of quadratic forms). Let y ∼CN (0M , IM ). Let A be an M ×M matrix independent ofy, which has a bounded spectral norm; that is, there existsCA < ∞ such that ‖A‖2 ≤ CA. Then, for any p ≥ 1, thereexists a constant Cp depending only on p, such that

Ey

[∣∣∣∣1

MyHAy − 1

Mtr(A)

∣∣∣∣p]≤ CpC

pA

Mp/2, (76)

By choosing p ≥ 2, we thus have that

1

MyHAy − 1

Mtr(A)→ 0. (77)

The following lemma provides results allowing to approx-imate random quantities involving the resolvent matrix whentheir dimensions grow simultaneously large:

Lemma 8. Let δ(t) be the unique positive solution to thefollowing equation:

δ(t) =M

K

(1 + t

K

K∑i=1

αi1+tδ(t)αi

) . (78)

Consider the asymptotic regime in which M and K grow toinfinity with: 0 < lim inf MK < lim sup M

K <∞. Let [a, b] be a

closed bounded interval in [0,∞). the following convergencesholds true:

supt∈[a,b]

∣∣∣∣1

Ktr Q(t)− δ(t)

∣∣∣∣→ 0. (79)

Moreover, if y1, · · · ,yK denotes standard complex Gaussianvectors independent from x1, · · · ,xK , we have:

maxj

supt∈[a,b]

∣∣yHj Q(t)yj − δ(t)∣∣→ 0. (80)

Note that, as a consequence of the rank-one perturbationlemma, the above convergences can be transferred to theresolvent matrix Qk(t). As a matter of fact, we also have:

supt∈[a,b]

∣∣∣∣1

Ktr Qk(t)− δ(t)

∣∣∣∣→ 0. (81)

and

maxj

supt∈[a,b]

∣∣yHj Q(t)yj − δ(t)∣∣→ 0. (82)

APPENDIX BPROOF OF THEOREM 1

We aim at determining deterministic equivalents of {qk}.Let zk = β

− 12

k hk. Define Qk = (∑m6=k

q`K h`h

H` + 1

ρIM )−1.Then, qk writes as:

hk =γk τ

βkK zHk Qkzk

Intuitively, from rank-one perturbation arguments (See Lemma6), all dk , zHk Qkzk present the same asymptotic behaviorand should converge to the same limit. In light of this obser-vation, we will rather focus on the study of the convergenceof {dk}. The convergence of {qk} to {qk} will then follow.

It can be thus easily shown that {dk} are the positivesolutions to the following fixed-point equations:

dk =1

KzHk

∑

m 6=k

τ γm

KdmzmzHm +

1

ρIM

−1

zk. (83)

Note that direct application of standard random matrix theorytools to the quadratic form arising in the expressions of {dk}is not analytically correct since coefficients {dk} and τ areboth function of the channel vectors {zk}. However, onewould expect coefficients {dm}m 6=k to be weakly dependent ofzk, and thus considering {dk} as deterministic, although notproperly correct, would lead to infer about their asymptoticbehavior. Based on these intuitive arguments and using theresults of Lemma 7 and Lemma 8 (see (78)) when all dk arereplaced by the same quantity d, one could claim that {dk}must satisfy the following convergence:

maxk

∣∣dk/d− 1∣∣→ 0 (84)

where d is given by

d

ρ=M

K− 1

K

K∑

m=1

γmτ

1 + γmτ. (85)

10

It is worth mentioning that d constitutes an asymptotic random,not deterministic, equivalent of d as it depends on τ . Addi-tional work is needed to find a deterministic equivalent for d.This will be performed later. We will now focus on providinga rigorous proof for (84). To this end, we will make use ofthe approach developed in [29]. Let us define ek = dk/d, andassume without loss of generality that e1 < · · · < eK . We canthus write {dk} as:

dk =1

KzHk

∑

m 6=k

τ γmzmzHm

Kemd+

1

ρIM

−1

zk (86)

from which dividing by d we get:

ek =1

KzHk

∑

m 6=k

τ γmzmzHmKem

+d

ρIM

−1

zk. (87)

From monotonicity arguments, it follows that:

eK ≤1

KzHK

∑

m6=K

τ γmzmzHmKeK

+d

ρIM

−1

zK (88)

or, equivalently,

1 ≤ 1

KzHK

∑

m 6=k

τ γmzmzHmK

+deKρ

IM

−1

zK . (89)

To prove that maxk |ek − 1| → 0, we proceed by contradic-tion. Assume that there exists ` > 0 such that lim sup eK >1 + `. Then, eK is infinitely often larger than 1 + `. Let usrestrict ourselves to such a subsequence. Therefore, we have:

1 ≤ 1

KzHK

τ

∑

m6=k

γmzmzHmK

+d(1 + `)

ρIM

−1

zK . (90)

We shall now invoke the convergence results of Lemma 8 inAppendix A. But before that, we need to check that ρτ/dstay almost surely in a bounded interval. This can be shownby noticing that function f can be bounded above and belowby:

MK

xρ + 1

≤ f(x) ≤ Mρ

Kx(91)

and thus:

ρ

(M

K− 1

)≤ d ≤ Mρ

K. (92)

Now, we are ready to apply the results of Lemma 8 to obtain:∣∣∣∣∣∣∣

1

KzHK

τ

∑

m 6=k

γmzmzHmK

+d(1 + `)

ρIM

−1

zK − µ

∣∣∣∣∣∣∣→ 0

(93)

where µ is the unique solution to the following equation:

µ =M

K

(d(1 + `)

ρ+

1

K

K∑

m=1

γmτ

1 + γm1+γmµτ

)−1

. (94)

The above convergence along with (90) implies that:

1 ≤ 1

KzHK

τ

∑

m6=k

γmzmzHmK

+d(1 + `)

ρIM

−1

zK

≤ µ+ εM (95)

for εM → 0. Observe that µ = f(d(1 + `)/ρ). Since f(d) = 1and f is decreasing, 1 − εM ≤ µ = f(d(1 + `)/ρ) < 1.Therefore, a contradiction arises when n tends to infinity. Thisproves that lim sup eK ≤ 1 for all large K. Using similararguments, we can prove that lim inf e1 ≥ 1. Plugging theseresults together, we finally obtain (84). Note that d is stillrandom because of its dependence on τ . Further work isneeded to find a deterministic equivalent for τ . Recalling thatqk = γk

βkτ

dk, 1K

∑Kk=1 qk = Pmax, and using (17), we obtain

τ =Pmax

1K

∑Kk=1

γkdkβk

(96)

Using (84), we thus have:

τ =Pmax

1K

∑Kk=1

γkdβk

+ o(1). (97)

where o(1) denotes a sequence converging to zero almostsurely. Replacing d by τ

K

∑Kk=1

γkβkPmax

, we finally get that:

τ =M

K

(α+

1

K

K∑

m=1

γm1 + τ γm

)−1

+ o(1). (98)

with α = 1K

∑K`=1

γ`ρβ`Pmax

. Using the above equation, weare tempted to discard the vanishing terms and to state that adeterministic equivalent by τ is given by τ , the unique solutionto the following equation:

τ =M

K

(α+

1

K

K∑

m=1

γm1 + τγm

)−1

. (99)

This is indeed true, since straightforward calculations lead tothe following identity:

τ − τ = o(1) +M

K

1K

K∑m=1

γ2m

(1+τγm)(1+τγm)

(α+ 1

K

K∑j=1

γj1+τγj

)(α+ 1

K

K∑j=1

γj1+τγj

) .

Using again the expressions of τ and τ , we have:

τ − τ = o(1) + τ τK

M

( 1

K

K∑

m=1

γ2m (τ − τ)

(1 + τγm)(1 + τ γm)

). (100)

Hence,

|τ − τ | ≤ K

M|τ − τ |+ o(1) (101)

from which it follows τ − τ → 0. Let d = τP

1K

∑Kk=1

γkβk

, wethus have maxk |dk − d| → 0. Putting the convergence resultsof τ and {dk} together, the convergence of qk directly follows.

11

APPENDIX CPROOF OF LEMMA 3

The aim of this section is to prove the almost sure con-vergence of {SINRdl

k } to {SINRdl

k }. First, using standardcalculations from random matrix theory, mainly the resolventlemma [30], it can be shown that asymptotic behavior of{SINRdl

k } remains almost surely the same if we replace pkby pk and qk by qk. This brings us to study the asymptoticexpression of the following quantity:

pkK

∣∣hHk uk∣∣2

∑i6=k

piK

∣∣hHk ui∣∣2 + 1

ρ

(102)

with ui = vi/‖vi‖ and vi given by (26). Denote bySk = pk

K

∣∣hHk uk∣∣2 and Ik =

∑i6=k

piK

∣∣hHk ui∣∣2 the signal and

interference terms, respectively. Let

Q(ρ) =

1

K

K∑

j=1

qjhjhHj +

1

ρIM

−1

(103)

denote the resolvent matrix associated with 1K

∑Kj=1 qjhjh

Hj .

For i ∈ {1, · · · ,K}, denote by Qi(ρ) the resolvent matrixobtained by removing the contribution of hi:

Qi(ρ) =

1

K

∑

j 6=iqjhjh

Hj +

1

ρIM

−1

(104)

Since Q(ρ)hi

‖Q(ρ)hi‖ = Qi(ρ)hi

‖Qi(ρ)hi‖ , then Sk can be written as:

Sk = pk

∣∣∣ 1KhHk Qk(ρ)hk

∣∣∣2

1K hHk Q2

k(ρ)hk. (105)

Applying Lemma 8 in Appendix A, it can be proved that:

maxk

∣∣∣ ρK

hHk Qk(ρ)hk − βk√

1− η2µ∣∣∣→ 0 (106)

where µ is the solution of:

µ =M

K

(1

ρ+

1

K

K∑

i=1

qiβi1 + qiβiµ

)−1

. (107)

To handle the denominator of Sk in (105), observe that:

maxk

∣∣∣∣1

KhHk Q2

k(ρ)hk −βkK

tr Q2(ρ)

∣∣∣∣→ 0, (108)

which is a consequence of the asymptotic properties ofquadratic forms and the rank-one perturbation in Lemma 1.Now, applying the results of [31, Proposition 3], we obtain:

1

Ktr Q2(ρ)− µ2

MK − µ2M

K1K

K∑i=1

β2i q

2i

(1+µβiqi)2

→ 0. (109)

From Theorem 1, it is clear that µ = γkβk

τqk

. Using this relationinto the above equation, we obtain:

maxk

∣∣∣∣1

KhHk Q2

k(ρ)hk − βkµ∣∣∣∣→ 0, (110)

where:

µ =µ2

MK − M

K1K

K∑i=1

(γiτ)2

(1+γiτ)2

. (111)

Putting all these results together yields the following conver-gence:

maxk

∣∣∣∣Sk −pkβk(1− η2)µ2

µ

∣∣∣∣→ 0. (112)

We now proceed to computing the interference term, whichcan be written as:

Ik =1

K2hHk Q(ρ)HkDkH

Hk Q(ρ)hk (113)

where Dk is a K − 1×K − 1 diagonal matrix given by:

Dk = diag

{p1

1K hH1 Q2(ρ)h1

, · · · , pk−1

1K hHk−1Q

2(ρ)hk−1

,

pk+1

1K hHk+1Q

2(ρ)hk+1

, · · · , pK1K hHKQ2(ρ)hK

}. (114)

Using the fact that:

hHk Q2(ρ)hk =hHk Q2

k(ρ)hk(1 + ρqk

1K hHk Qk(ρ)hk

)2 (115)

and exploiting the already established convergences in (106)and (110), we can prove that:

maxk

∣∣∣∣∣pk

1K hHk Q2(ρ)hk

− pk(1 + qkβkµ)2

βkµ

∣∣∣∣∣→ 0. (116)

It entails from the above convergence that matrix Dk con-verges in operator norm to Dk obtained by replacing therandom elements of Dk by their asymptotic equivalents.Studying the asymptotic behaviour of Ik amounts thus toconsidering Ik given by:

Ik =ρ

K2hHk Q(ρ)HkDkH

Hk Q(ρ)hk. (117)

Using the decomposition of Q as:

Q(ρ) = Qk(ρ)−1K qkQk(ρ)hkh

Hk Qk(ρ)

1 + 1K qkh

Hk Qk(ρ)hk

(118)

we can expand Ik as:

Ik =1

K2hHk Qk(ρ)HkDkH

Hk Qk(ρ)hk

− qkK3

hHk Qk(ρ)hkhHk Qk(ρ)HkDkHkQk(ρ)hk

(1 + qkK hHk Qkhk)

− qkK3

hHk Qk(ρ)HkDkHHk Qk(ρ)hkh

Hk Qk(ρ)hk

(1 + qkK hHk Qkh)

+q2k

K4

∣∣∣hHk Qk(ρ)hk

∣∣∣2

hHk Qk(ρ)HkDkHHk hk

(1 + qk

K hHk Qk(ρ)hk

)2 . (119)

12

This expression makes arise classical quadratic forms that canbe studied using the trace Lemma. We thus have:

Ik =1

K2βk tr Qk(ρ)HkDkH

Hk Qk(ρ)

− qkβ2k(1− η2)µ

1 + qkβkµ

1

K2tr Qk(ρ)HkDkH

Hk Qk(ρ)

− qkβ2k(1− η2)µ

1 + qkβkµ

1

K2tr Qk(ρ)HkDkH

Hk Qk(ρ)

+q2k(1− η2)µ2β3

k

(1 + qkβkµ)2

1

K2tr Qk(ρ)HkDkH

Hk Qk(ρ) + εk

= µkβkK2

tr Qk(ρ)HkDkHHk Qk(ρ) + εk (120)

where εk is a random sequence converging to zero almostsurely uniformly in k, that is maxk |εk| → 0, and

µk =1 + 2η2βkqkµ+ η2µ2q2

kβ2k

(1 + qkβkµ)2(121)

=1 + 2η2γkτ + η2γ2

kτ2

(1 + qkβkµ)2. (122)

The second equality is obtained using the factthat µqkβk = γkτ . We will now handle the term

1K2 tr Qk(ρ)HkDHH

k Qk(ρ). Note that due to the rank-one perturbation Lemma and the convergence in operatornorm of Dk to Dk, the matrices Qk(ρ) and Dk can bereplaced by Q(ρ) and Dk. In doing so, we prove that Ik isalmost surely equivalent to:

µkβkK2

tr Q(ρ)HkDHHk Q(ρ) (123)

= µkβkK2

K∑

i=1,i6=k

hHi Q(ρ)2hipi1K hHi Q(ρ)2hi

= µkβkK

K∑

i=1,i6=kpi. (124)

Since 1K

∑Ki=1,i6=k pi = Pmax +O(1/K), we thus have:

maxk|Ik − µkρβkPmax| → 0. (125)

Putting the above results together yields Lemma 3.

APPENDIX DPROOF OF LEMMA 5

In this proof, we compute deterministic equivalents of theentries of the matrices ak, Ek and Bk,i. We start introducingthe following functionals:

Xk(t) =1

KhHk Q(t)hk (126)

Yk(t) =1

KhHk Q(t)hk. (127)

The coefficients of ak and Ek can be written as a function ofthe higher derivatives of the above functionals taken at t = 0as follows:

[ak]` =(−1)`

`!Y

(`)k (128)

[Ek]`,m =(−1)`+m

`!m!X

(`+m)k . (129)

Thus, we need to compute deterministic equivalents of Y (`)k

and X(`)k . In [17], it has been shown that it suffices to

determine deterministic equivalents of Xk(t) and Yk(t) andthen take their derivatives at t = 0. We begin first by treatingX

(`)k . Using Lemma 6, we can write

1

KhHk Q(t)hk =

1K hHk Qk(t)hk

1 + tqkK hHk Qk(t)hk

. (130)

Lemma 7 along with the rank-one perturbation property inLemma 6 implies that

1

KhHk Qk(t)hk −

1

Kβk tr (Q(t))→ 0. (131)

Using Lemma 8, we can conclude that1

KhHk Qk(t)hk − βkδ(t)→ 0. (132)

Then, Xk(t)−Xk(t)→ 0 where Xk(t) is defined as in (52).Using Corollary 6 in [17], we have X(`)

k − X(`)

k → 0 suchthat:

wHk aka

Hk wk −wH

k akaHk wk → 0. (133)

Again using Lemma 6, we can write

1

KhHk Q(t)hk =

1KhHk Qk(t)hk

1 + tqkK hHk Qk(t)hk

. (134)

The asymptotic equivalent of the quadratic form1KhHk Qk(t)hk, is the same as

√1−η2K hHk Qk(t)hk. Thus,

Yk(t)− Y k(t)→ 0 where

Y k(t) =

√1− η2βkδ(t)

1 + tqkβkδ(t). (135)

Using again Corollary 6 in [17], we have Y (`)k − Y (`)

k → 0such that

wHk Ekwk −wH

k Ekwk → 0. (136)

We are thus left with studying the convergence of the inter-ference term:

∑

i 6=k

piK

wHi Bk,iwi =

J−1∑

`=0

J−1∑

m=0

∑

i 6=k

piKw`,iwm,i[Bk,i]`,m (137)

Let Dk = diag(p1w`,1wm,1, · · · , pk−1w`,k−1wm,k−1, pk+1

w`,k+1wm,k+1, · · · , pKw`,Kwm,K) and rewrite each term∑i 6=k

piKw`,iwm,i[Bk,i]`,m as

1

K2hHk

(HQHH

K

)`HkDkH

Hk

(HQHH

K

)mhk (138)

=(−1)`+m

m!`!Z

(`,m)k (139)

where Z(`,m)k is the (`,m) derivative taken on t = 0 and u = 0

of the functional Zk(t, u) defined as

Zk(t, u) =1

K2hHk Q(t)HkDkH

Hk Q(u)hk. (140)

Using same techniques as in Appendix C, one can prove that

Zk(t, u) = fk(t, u)1

K2tr Qk(t)HkDkH

Hk Qk(u) + εk (141)

13

with fk(t, u) given in (54) and εk is such that maxk |εk| → 0.Using the same techniques as in [17, Lemma 15], one canprove that

1

K2tr Qk(t)HkDkH

Hk Qk(u)−

α(t, u)1

K

∑

i 6=k

piw`,iwm,iβi(1 + tδ(t)βiqi)(1 + uδ(u)βiqi)

→ 0 (142)

where

α(t, u) =δ(t)δ(u)

MK − tu

K δ(t)δ(u)K∑i=1

[βiqi]2

[1+tqiβiδ(t)][1+uqiβiδ(u)]

.

Therefore, we have that Zk(t, u) − 1K

∑i 6=k Zk,i(t, u) → 0

with Zk,i(t, u) given by (53). Also,∑

i 6=k

piK

wHi Bk,iwi −

∑

i 6=k

piK

wHi Bk,iwi → 0 (143)

where [Bk,i]`,m is given in (57). Similarly, it can be shownthat

∑

i 6=k

qiK

wHk Bi,kwk −

∑

i6=k

qiK

wHk Bi,kwk → 0, (144)

Plugging all these results together yields the convergenceresults of Lemma 5.

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